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a(n) = [x^n](1/2)*(1 + (2*x + 1)/sqrt(1 - 8*x^2*(x + 1))).
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%I #6 Feb 28 2023 03:54:49

%S 1,1,2,6,16,48,140,424,1280,3920,12032,37184,115248,358624,1118784,

%T 3499584,10969344,34450944,108377984,341465344,1077300224,3403006464,

%U 10761447424,34065967104,107937899264,342293526016,1086339120128,3450236511232,10965437349888

%N a(n) = [x^n](1/2)*(1 + (2*x + 1)/sqrt(1 - 8*x^2*(x + 1))).

%F a(n) = (4*(2*n^2 - 11*n + 15)*a(n - 3) + 4*(2*n^2 - 9*n + 9)*a(n - 2) + 2*(n - 3)*a(n - 1)) / (n^2 - 3*n) for n >= 4.

%p gf := (1/2)*(1 + (2*x + 1)/sqrt(1 - 8*x^2*(x + 1)));

%p ser := series(gf, x, 30): seq(coeff(ser, x, n), n = 0..28);

%p # Recurrence:

%p a := proc(n) option remember; if n < 4 then return [1, 1, 2, 6][n + 1] fi:

%p (4*(2*n^2 - 11*n + 15)*a(n - 3) + 4*(2*n^2 - 9*n + 9)*a(n - 2) + 2*(n - 3)*a(n - 1)) / (n^2 - 3*n) end: seq(a(n), n = 0..28);

%Y Cf. A360571.

%K nonn

%O 0,3

%A _Peter Luschny_, Feb 28 2023